I have exhausted my state extended unemployment but am still looking for work. I heard there was more federal unemployment available. Is there any way to get at least partial compensation if I can only find a part time job?
What you heard was that a federally funded extension of your state unemployment benefits was available. It is not federal unemployment, and it is administered by the same state agency as your original benefits. The only difference is that the state agency gets the money from the federal government instead of from the state government.
In most states, you can collect partial benefits when working part time and earning less than a certain amount per week. However, the rules vary. For example, NY decided that it would pay partial benefits only if you work 3 or fewer days per week.
The Federal extended benefits are paid through the state. You need to continue to report to the state just like you were before you exhausted those benefits.